Long paths in hypercubes with a quadratic number of faults

Long cycles in hypercubes with optimal number of faulty vertices

“…Moreover, they [4] presented a function φ(n) = Θ(n 6 ) such that the problem remains NP-hard even if |F | ≤ φ(n). Furthermore, Dvořák and Koubek [3] described a polynomial algorithm for the similar decision problem of long F -free paths between given vertices in Q n if |F | ≤ n 2 /10 + n/2 + 1.…”

“…In the first part, we consider the case when Q n can be split into Q i:L and (3). In this case, we apply induction directly.…”

“…In the first part of the proof of Theorem 1.1 we assume that there exists a dimension i ∈ [n] such that |F i:L |, |F i:R | ≤ n−1 2 − 2; see (3). In this case we apply induction (2) in both Q i:L and Q i:R to construct a long F -free cycle Q n .…”

Conditional edge-fault Hamiltonicity of augmented cubes☆

“…Many embedding problems in various faulty networks have been solved [3,6,9,10,[12][13][14][15][16][18][19][20][21][22][23][24][25][28][29][30]37,38,40,43,44]. Specifically, the edge-fault Hamiltonicity [13][14][15]18,22,28,34,[36][37][38] investigates the Hamiltonicity of networks under two major faults assumptions: (1) the random fault-assumption, which posits that there is no restriction on the distribution of faulty edges [13,18,22,34,38]; and (2) the conditional fault-assumption, which posits that each node is incident to at least two fault-free edges [14,15,28,36].…”

Edge-fault-tolerant diameter and bipanconnectivity of hypercubes

“…For (fault-tolerant) path and/or cycle embedding into hypercubes, see recent Refs. [2][3][4][5][6][7][8][9][10][11][13][14][15]17,[19][20][21][22] and a survey [23].…”